1. Field of Invention
The present invention is related to a method and apparatus for generating a symmetric-periodic Continuous Phase Modulation (CPM) signal; and more particularly, is related to a method and apparatus for generating a symmetric-periodic CPM signal in a high speed wireless packet network such as that set forth in the IEEE 802.16e Standard for wireless Metropolitan Area Network (MAN) technology.
2. Description of Related Art
Orthogonal Frequency Division Multiplexing (OFDM) transmission schemes are well known in the art for transmitting data in broadband multi-user communications systems and network, as well as other known systems and networks, and was first introduced as a means of counteracting channel-induced linear distortions encountered when transmitting over a dispersive radio channel. See L. Hanzo, et al., “OFDM and MC-CDMA for Broadband Multi-User Communications, WLANs and Broadcasting,” J. Wiley & Sons, Ltd., 2004; as well as A. Bahai et al., “Multi-Carrier Digital Communications Theory and Applications of OFDM”, 2nd Ed., Springer Science and Business, Inc. 2004.
For such OFDM transmission schemes, inter-symbol interference (ISI) and inter-carrier interference (ICI) can be removed at the receiver by adding a cyclic guard interval and a cyclic prefix to the time-domain transmitted signal. This is accomplished by pre-pending a certain number of the ending data vector to the beginning of the OFDM symbol (or, equivalently, by appending a certain number of the beginning data vector to the end of the OFDM symbol). If the guard interval is longer in duration than the channel's impulse response, then each sub-carrier will appear to have passed through a flat fading channel. Consequently, the receiver can exploit the cyclic shift properties of the Discrete Fourier Transform (DFT) to significantly reduce the complexity of frequency domain equalization (FDE) techniques.
For example, FIG. 1A shows blocks of data 6, 8 having cyclic extensions 10, 12 postfixed thereon in relation to corresponding blocks of data 13, 15 having cyclic extensions 14, 16 prefixed thereon. When transmitted, each block of data is linearly convolved with the channel. By adding the cyclic extension (prefix or postfix) to each block, one can make the linear convolution between the block and the channel appear to be a circular convolution if the length of the guard interval exceeds the impulse response length of the channel. In the frequency domain, one can implement a single-tap channel equalizer at each frequency. This technique is well known for OFDM-based communications networks and systems and, more recently, for single-carrier systems. It has only recently been considered for CPM-based applications. In FIG. 1A, there is a window (L . . . G) over which the FFT window may start. As long an Nk-point FFT is taken (N data symbols/block and k samples/symbol), one can obtain an equivalent receiver output.
Moreover, DFT-based SC-FDE (Single-Carrier FDE) techniques have only recently been applied to Continuous Phase Modulation (CPM) systems. For the purpose of understanding the invention that is discussed herein, CPM is summarized and characterized as follows: Over the nth symbol interval, a binary single-h CPM waveform can be expressed as
                                          s            ⁡                          (                              t                ,                a                ,                h                            )                                =                      exp            ⁢                          {                              j                ⁢                                                                  ⁢                2                ⁢                                                                  ⁢                π                ⁢                                                                  ⁢                h                ⁢                                                      ∑                                          i                      =                                              -                        ∞                                                              n                                    ⁢                                                            I                      i                                        ⁢                                          q                      ⁡                                              (                                                  t                          -                                                      i                            ⁢                                                                                                                  ⁢                            T                                                                          )                                                                                                        }                                      ,                  nT          ≤          t          <                                    (                              n                +                1                            )                        ⁢            T                          ,                            (        1        )            where T denotes the symbol duration, Iiε{±1} are the binary data bits and h is the modulation index. The phase function, q(t), is the integral of the frequency function, f(t), which is zero outside of the time interval (0,LT) and which is scaled such that
                                          ∫            0            LT                    ⁢                                    f              ⁡                              (                τ                )                                      ⁢                          ⅆ              τ                                      =                              q            ⁡                          (              LT              )                                =                                    1              2                        .                                              (        2        )            An M-ary single-h CPM waveform is the logical extension of the binary single-h case in which the information symbols are now multi-level: i.e., Iiε{±1, ±3, . . . , ±(M−1)}. Usually, M is selected to be an even number. However, it is noted that other alphabets are possible (and can also be used with this invention). For example, M can be odd or the alphabet can include zero—i.e. Iiε{0, ±1, ±3, . . . , ±(M−1)}. The only restriction in this invention is that the alphabet contains an element and its antipodal counterpart. Finally, an M-ary multi-h CPM waveform can be written as
                                          s            ⁡                          (                              t                ,                I                ,                h                            )                                =                      exp            ⁢                          {                              j2π                ⁢                                                                  ⁢                                                      ∑                                          i                      =                                              -                        ∞                                                              n                                    ⁢                                                            I                      i                                        ⁢                                          h                                                                        (                          i                          )                                                J                                                              ⁢                                          q                      ⁡                                              (                                                  t                          -                          iT                                                )                                                                                                        }                                      ,                  nT          ≤          t          <                                    (                              n                +                1                            )                        ⁢                          T              .                                                          (        3        )            Typically, Iiε{±1, ±3, . . . , ±(M−1)} (M even). However, there is no restriction to this particular alphabet and M can even be odd, as mentioned earlier. Typically, the modulation index cycles through a set of J values: hε{h0ΛhJ−1} and so (i)J denotes “i modulus J”. The expression in (3) may also be written as:
                              s          ⁡                      (                          t              ,              a              ,              h                        )                          =                  exp          ⁢                      {                          j              (                                                θ                                      n                    -                    L                                                  +                                  2                  ⁢                                                                          ⁢                  π                  ⁢                                                                          ⁢                                                            ∑                                              i                        =                        0                                                                    L                        -                        1                                                              ⁢                                                                  I                                                  n                          -                          i                                                                    ⁢                                              h                                                                              (                                                          n                              -                              i                                                        )                                                    J                                                                    ⁢                                              q                        ⁡                                                  (                                                      t                            -                                                                                          (                                                                  n                                  -                                  i                                                                )                                                            ⁢                              T                                                                                )                                                                                                                                }                                                          (        4        )            The phase state,
      θ          n      -      L        =      π    ⁢                  ∑                  i          =                      -            ∞                                    n          -          L                    ⁢                        I          i                ⁢                  h                                    (              i              )                        J                              mod 2π determines the contribution of the symbols for which the phase function has reached its final constant value of one half.
However, when applying such DFT-based SC-FDE techniques to CPM systems, some issues have developed. Since the CPM waveform signal is supposed to have a continuous phase, one cannot simply append a cyclic extension at the end or beginning of a data block. FIG. 1B shows an example of a blind introduction of a cyclic extension, which can destroy the continuous phase property of the CPM waveform signal. If the cyclic postfix portion of the waveform is appended to a CPM waveform, the phase would become discontinuous, which results in expansion of the signal bandwidth and a reduction in spectral efficiency. In effect, when pre-pending or appending the cyclic extension to the CPM waveform, care must be taken in order to maintain phase continuity.
One approach for appending a cyclic extension to CPM block transmissions is to insert special data-dependent symbols (“channel” or “tail” symbols) into the data portion of the CPM transmission block. The inclusion of these special symbols allows the transmitter to repeat the data in a cyclic extension without destroying the continuous phase property of the signal. However, these “channel” symbols, which are calculated based on past observations, must either be computed on a block-by-block basis or determined by using a table-lookup in order to map a particular sequence of observed symbols to the required “channel” symbol sequence. In addition, since they are data-dependent, the actual number of “channel” bits that are needed may vary from block to block. Simple approaches exist for constructing the “tail” bits for binary single-h CPM systems, but no one has provided a general, low complexity solution for M-ary multi-h CPM.
Recently, DTT-based (Discrete Trigonometric Transform) SC-FDE techniques have been applied to OFDM as an alternative to DFT-based FDE schemes. See Giri Mandyam, “Sinusoidal Transforms in OFDM Systems”, IEEE Transactions in Broadcasting, Vol. 50, No. 2, June 2004, pp. 172-184. DTT includes the family of Discrete Sine Transforms (DSTs) and Discrete Cosine Transforms (DCTs). Just as DFT-based system exploit the cyclic convolution property of the DFT in order to simplify receiver design, DTT-based systems can exploit the symmetric-convolution property of the DTT in order to develop low complexity FDE techniques.
The symmetric convolution property of the DTT can be summarized as follows: the cyclic convolution of two symmetrically extended finite sequences in the time domain is equivalent to the multiplication of their cosine/sine series coefficients in the frequency domain. Thus, in order to use DTT-based receiver methods, a transmitter can create a signal that has symmetry/anti-symmetry about a distinct point in time, and which repeats in a cyclic prefix/postfix. When passed through the radio channel, it will appear to have passed through a flat fading channel and consequently the receiver can exploit the symmetric-convolution properties of the DTT to significantly reduce the complexity of FDE techniques, in the same manner as the DFT has been used with waveforms that have been extended to transmit a cyclic prefix/postfix.
However, DTT-based FDE methods have not been applied to CPM because the concept of symmetric-periodic CPM does not exist.
Although it is relative straightforward to create a symmetrical extension to a linearly modulated full-response (i.e. memory-less) signal, it is actually quite challenging to do the same for CPM because the output waveform is a nonlinear function of the input symbols and because CPM systems have memory. This means that the waveform which is observed over a particular symbol interval is dependent on the current state of the system, which is a function of past symbols. Thus, in order to create special properties in the observed waveform (such as symmetry and periodicity), the transmitter must take the system memory into account and construct additional input symbols that will create the desired signal properties while preserving the continuous phase/constant envelope characteristics that make CPM so attractive. In fact, to date, the only low complexity FDE techniques that have been developed for use with CPM have been based on the construction of a cyclic extension to CPM (cyclic prefix). This generally requires the calculation of additional, input symbols as a function of the past information symbols in order to create the cyclic extension without destroying the continuous phase property of CPM, and the prior art (for the creation of cyclic extensions) has only focused on the binary single-h case, which is the least complex scenario.
The technique of creating a symmetric-periodic OFDM waveform for use with DTT-based FDE methods was first introduced in Giri Mandyam, “Sinusoidal Transforms in OFDM Systems”, IEEE Transactions in Broadcasting, Vol. 50, No. 2, June 2004, pp. 172-184, where it was found that when the duration of the channel exceeded the length of the OFDM guard interval, that use of symmetric-periodic OFDM outperformed the use of cyclically extended OFDM due to the extra redundancy in the transmitted signal. In addition, when the received signal has been corrupted by bursty interference that is present over a portion of the block, the redundancy may be useful for interference cancellation.
However, there is no known prior art for creating a symmetric-periodic CPM waveform.